Abstract view
Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space


Published:20130820
Printed: Oct 2014
Lucijan Plevnik,
Institute of Mathematics, Physics, and Mechanics, Jadranska 19, SI1000 Ljubljana, Slovenia
Peter Šemrl,
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI1000 Ljubljana, Slovenia
Abstract
Let $\mathcal{H}$ and $\mathcal{K}$ be infinitedimensional separable
Hilbert spaces and ${\rm Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$.
We describe the general form of pairs of bijective maps $\phi , \psi :
{\rm Lat}\,\mathcal{H} \to {\rm Lat}\,\mathcal{K}$ having the property that for every pair
$U,V \in {\rm Lat}\,\mathcal{H}$ we have $\mathcal{H} = U \oplus V \iff \mathcal{K} = \phi (U) \oplus \psi (V)$. Then we reformulate this theorem as a description
of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known
structural results for maps on idempotents are easy consequences.